Mercurial > hg > ltpda

annotate m-toolbox/html_help/help/ug/franklin_ng_content.html @ 51:9d5c88356247 database-connection-manager

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Make unit tests database connection parameters configurable
author Daniele Nicolodi <nicolodi@science.unitn.it>
date Wed, 07 Dec 2011 17:24:37 +0100
parents f0afece42f48
children
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
0
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
1 The following sections gives an introduction to the <a href="noisegen.html">generation of model noise</a> using the noise generator implemented in LTPDA.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
2 <ul>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
3 <li><a href="#franklin">Franklin's noise generator</a></li>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
4 <li><a href="#description">Description</a></li>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
5 <li><a href="#inputs">Inputs</a></li>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
6 <li><a href="#outputs">Outputs</a></li>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
7 <li><a href="#usage">Usage</a></li>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
8 </ul>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
9 <h2><a name="franklin">Franklin's noise generator</a></h2>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
10 Franklin's noise generator is a method to generate arbitrarily long time series with a prescribed spectral density.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
11 The algorithm is based on the following paper:
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
12 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
13 <p>Franklin, Joel N.:
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
14 <i> Numerical simulation of stationary and non-stationary gaussian
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
15 random processes </i>, SIAM review, Volume {<b> 7</b>}, Issue 1, page 68--80, 1965.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
16 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
17 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
18 The Document <i> Generation of Random time series with prescribed spectra </i> by Gerhard Heinzel (S2-AEI-TN-3034) <br> corrects a mistake in the aforesaid paper and describes the practical implementation.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
19 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
20 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
21 See <a href="noisegen.html">Generating model noise</a> for more general information on this.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
22 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
23 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
24 Franklin's method does not require any 'warm up' period. It starts with a transfer function given as ratio of two polynomials.<br/>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
25 The generator operates on a real state vector y of length n which is
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
26 maintained between invocations. It produces samples of the time series in equidistant steps <tt>T = 1/fs</tt>, where <tt>fs</tt> is the sampling frequency.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
27 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
28 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
29 <ul>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
30 <li> y0 = Tinit * r, on initialization
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
31 <li> yi = E * yi-1 + Tprop * r, to propagate
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
32 <li> xi = a * yi , the sampled time series.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
33 </ul>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
34 r is a vector of independent normal Gaussian random numbers
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
35 Tinit, E, Tprop which are real matrices and a which is a real vector are determined once by the algorithm.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
36 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
37
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
38 <h2><a name="description">Description</a></h2>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
39 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
40 When an analysis object is constructed from a pole zero model Franklin's noise generator is called (compare <a href="ao_create.html#pzmodel">Creating AOs from pole zero models</a>).
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
41 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
42
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
43
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
44 <h2><a name="inputs">Inputs</a></h2>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
45 for the function call the parameter list has to contain at least:
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
46 <ul>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
47 <li> nsecs - number of seconds (length of time series)
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
48 <li> fs - sampling frequency
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
49 <li> pzmodel with gain
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
50 </ul>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
51
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
52 <h2><a name="outputs">Outputs</a></h2>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
53 <ul>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
54 <li> b - analysis object containing the resulting time series
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
55 </ul>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
56 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
57 <h2><a name="usage">Usage</a></h2>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
58 The analysis object constructor <a href="ao_create.html">ao</a> calls the following four functions when the input is a pzmodel.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
59 <ul>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
60 <li> ngconv
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
61 <li> ngsetup
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
62 <li> nginit
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
63 <li> ngprop
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
64 </ul>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
65 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
66 First a parameter list of the input parameters is to be done. For further information on this look at <a href="plist_create.html#params">Creating parameter lists from parameters</a>.<br/>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
67 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
68 <h2><a name="starting">Starting from a given pole/zero model</a></h2>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
69 <p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
70 The parameter list should contain the number of seconds the resulting time series should have <tt>nsecs</tt> and the sampling frequency <tt>fs</tt>. <br/>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
71 The constructor call should look like this:
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
72 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
73 <div class="fragment"><pre>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
74 f1 = 5;
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
75 f2 = 10;
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
76 f3 = 1;
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
77 gain = 1;
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
78 fs = 10; <span class="comment">% sampling frequency</span>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
79 nsecs = 100; <span class="comment">% number of seconds to be generated</span>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
80 p = [pz(f1) pz(f2)];
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
81 z = [pz(f3)];
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
82 pzm = pzmodel(gain, p, z);
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
83 a = ao(pzm, plist(<span class="string">'nsecs'</span>, nsecs, <span class="string">'fs'</span>,fs))
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
84
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
85 </pre></div>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
86 The output will be an analysis object <tt>a</tt> containing the time series with the spectrum described by the input pole-zero model.
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
87 </p>
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
88
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
89
f0afece42f48 Import.
Daniele Nicolodi <nicolodi@science.unitn.it>
parents:
diff changeset
90